Synchronous and asynchronous bursting states: role of ... - Springer Link
J Comput Neurosci (2007) 23:189–200 DOI 10.1007/s10827-007-0027-9
Synchronous and asynchronous bursting states: role of intrinsic neural dynamics Takashi Takekawa · Toshio Aoyagi · Tomoki Fukai
Received: 8 March 2006 / Revised: 28 December 2006 / Accepted: 20 February 2007 / Published online: 27 March 2007 © Springer Science + Business Media, LLC 2007
Abstract Brain signals such as local field potentials often display gamma-band oscillations (30–70 Hz) in a variety of cognitive tasks. These oscillatory activities possibly reflect synchronization of cell assemblies that are engaged in a cognitive function. A type of pyramidal neurons, i.e., chattering neurons, show fast rhythmic bursting (FRB) in the gamma frequency range, and may play an active role in generating the gamma-band oscillations in the cerebral cortex. Our previous phase response analyses have revealed that the synchronization between the coupled bursting neurons significantly depends on the bursting mode that is defined as the number of spikes in each burst. Namely, a network of neurons bursting through a Ca2+ -dependent mechanism exhibited sharp transitions between synchronous and asynchronous firing states when the neurons exchanged the bursting mode between singlet, doublet and so on. However, whether a broad class of bursting neuron models commonly show such a network behavior remains unclear. Here, we analyze the mechanism underlying this network behavior using a mathematically tractable neuron model. Then we extend our results to a multi-compartment version of the NaP current-based neuron model and prove a similar tight
Action Editor: Xiao-Jing Wang T. Takekawa (B) · T. Fukai Laboratory for Neural Circuit Theory, RIKEN Brain Science Institute, 2-1 Hirosawa, Wako, Saitama 351-0198 Japan e-mail: [email protected] T. Aoyagi Graduate School of Informatics, Kyoto University, Yoshida-Honmachi, Sakyo, 606-8501 Kyoto, Japan
relationship between the bursting mode changes and the network state changes in this model. Thus, the synchronization behavior couples tightly to the bursting mode in a wide class of networks of bursting neurons. Keywords Stimulus-dependent synchronization · Fast rhythmic burst · Phase response · Synaptic location
1 Introduction A type of pyramidal neurons called “fast rhythmic bursting (FRB)” or “chattering” neuron responds to a visual stimulus or a depolarizing step current with rhythmic bursts of spikes in the gamma frequency range (30–70 Hz) (Gray and McCormick 1996; Steriade et al. 1998; Cardin et al. 2005). These neurons are distinguished from other types of cortical neurons showing burst firing by their rather high-intraburst firing frequencies (> 200–300 Hz) (Gray and McCormick 1996), and may promote synchronization of neuronal activity in local cortical areas (Gray and Singer 1989; TallonBaudry et al. 1997; Gray and Prisco 1997; Singer 1999; Cunningham et al. 2004; Steriade 2006). The phase response curve (PRC) describes how a perturbative input to a neuron advances or delays a postsynaptic spike response in a rhythmically firing state (Kuramoto 1984; Ermentrout and Kopell 1984), and has recently been measured in cortical neurons (Reyes and Fetz 1993; Gutkin et al. 2005; Netoff et al. 2005; Tsubo et al. 2005; Goldberg et al. 2007). We have previously shown using PRCs that synchronization or desynchronization in a network of rhythmically bursting neurons might depend on the intrinsic mechanism
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of bursting. Some experimental evidence suggested that the calcium-activated non-specific cation (CAN) current (Wilson et al. 1996; Haj-Dahmane and Andrade 1997) is essential for FRB (Kang and Kayano 1994; Kang et al. 1998; Aoyagi et al. 2002). Other studies suggested that the persistent sodium (NaP) current and the electronic interplay between soma and dendrite are crucial for FRB (Wang 1999; Brumberg et al. 2000; Traub et al. 2003). Therefore we analyzed the synchronization properties of two types of neuron models when the same type of neurons were coupled via AMPA receptor-mediated excitatory synapses (Aoyagi et al. 2003; Takekawa et al. 2004). The coupled neuron system of the CAN current-based single-compartment model underwent an exchange of the stability from anti-phase (with π phase lag) to in-phase (with 0 phase lag) synchronous firing when the bursting mode, or the number of spikes per burst, increased (Aoyagi et al. 2003). By contrast, the NaP current-based twocompartment neuron model showed neither in-phase nor anti-phase phase-locked state in many cases, and the stability exchanges between them were also less evident (Takekawa et al. 2004). These results raise several questions about the relationship between the network dynamics for synchronization and the bursting dynamics which is intrinsic to cell types (Pinsky and Rinzel 1994; Izhikevich 2000; Booth and Bose 2002; Doiron et al. 2002). Are the correlated changes between the bursting mode and the stability of synchronous firing a general property of bursting neuron models? How does the intrinsic mechanism of bursting influence the network dynamics of synchronization? Here, we address these issues by introducing a coupled system of simple bursting neuron models (Izhikevich 2003). We then increase the number of dendritic compartments in the NaP current-based two-compartment neuron model to show that the tight relationship between the burst mode and synchronous firing holds for the extended model.
where x(t ± 0) = limε→0 x(t ± ε). Here, the values of the parameters were set as a = 0.02, b = 0.2, c = −50, d = 80, 50, 39, 28 (singlet, two types of doublet, and triplet), and I = 35. To make the sizes of u and v the same order, we have rescaled the variables in the original model as u ← u/b and d ← d/b . 2.2 Single-compartment CAN current model In the present paper, we analyzed two different realistic pyramidal neuron models that exhibit rhythmic bursting in the gamma frequency range (20–80 Hz): a single-compartment model based on Ca2+ -activated cation (CAN) current and a multi-compartment model based on NaP current. The CAN current-based model is described by the following equation for the membrane potential: Cm
dV = −Ileak − INa − IDK − ICa dt −IKCa − ICAN + Iext ,
(4)
where Ileak , INa and IDK are a leak current and spikegenerating sodium and potassium currents; ICa , IKCa and ICAN represent high-voltage-activated Ca2+ current, Ca2+ -activated potassium current (SK current) and Ca2+ -activated cation current, respectively; and Iext is an external input current. Since the mathematical details of this model have been published elsewhere (Aoyagi et al. 2002, 2003), below we list only the parameter values that were different from the previous ones: T = 309.16, KCAN = 13.16 μM and gCAN = 113 or 170 (two types of doublet) mS/cm2 . 2.3 Multi-compartment NaP current model
2 Materials and methods 2.1 Izhikevich model We used the bursting neuron model constructed by Izhikevich (2003). The model is described by twodimensional ordinary differential equations of the following form: dv = 0.04v + 5v 2 + 140 − b u + I ≡ Fv (u, v), dt du = a(v − u) ≡ Fu (u, v). dt
At each postsynaptic spike, the variables are reset as follows: v(t + 0) = c if v(t − 0) = 30 then (3) u(t + 0) = u(t − 0) + d,
(1) (2)
We extended the previously proposed twocompartment neuron model (Wang 1999). The present model has one somatic compartment and six dendritic compartments. The membrane potential of the n-th compartment Vn obeys the following equation:
Cm
dVn Vn − Vn−1 Vn − Vn+1 = − In − + dt Rn−1 An Rn An −
Iext,n + Isyn,n An
(n = 1, 2, · · · , 7),
(5)
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where n represents the serial position of the compartment from the soma which is indexed as n = 1. The membrane capacitance Cm is 1 μF/cm2 . The somatic compartment’s area A1 is 4950 μm2 and the dendritic compartment’s areas An (n = 2, · · · , 7) are fixed at 4675 μm2 . The axial resistance between adjacent compartments Rn (n = 1, · · · , 6) are fixed at 10 M. In 0 8 the above equations, we set as V1R−V = 0 and V7R−V =0 0 7 at both ends. The somatic compartment includes a leak current, the spike-generating sodium and potassium currents, and muscarine-dependent potassium current (M-current), while the dendritic compartments include a leak current, persistent sodium current, highvoltage-activated Ca2+ current, and Ca2+ -activated potassium current: I1 = Ileak + INa + IDK + IM ; In = Ileak + INaP + ICa + IKCa (n = 2, · · · , 7). These ionic currents are represented as Ileak = gleak (V − Eleak ), INa = gNa m3 h(V − ENa ), IK = gK m4 (V − EK ), IM = gM m(V − EK ), INaP = gNaP sig((V + 45)/5)(V − ENa ), ICa = gCa sig((V + 20)/10)(V − ECa ) and IKCa = gKCa [Ca2+ ]i (V − EK ), where [Ca2+ ] +30 i
sig(x) ≡
1 , 1 + exp(−x)
lin(x) ≡
x . 1 − exp(−x)
(6)
The activation (m) and inactivation (h) variables of each ionic current obey the following equation: dx/dt = α(1 − x) − βx. The coefficients α and β may depend on the membrane potential. The explicit forms of these functions are listed in Table 1 for the types of ionic current used in the present models. The concentration of intracellular calcium [Ca2+ ]i obeys the following differential equation: d[Ca2+ ]i = −0.002ICa − 0.005[Ca2+ ]i . dt
(7)
We investigated the neuronal responses when an external current was applied to different compartments. We consider the case that the input current is mediated by AMPA glutamate receptors: Iext = gext (V − EAMPA ). We change the intensity of the input current by changing the value of gext . Table 1 The coefficent functions of the activation and inactivation variables for the ionic currents of the NaP current model Type
α
β
Na, m Na, h DK, m M, m
10 lin((V + 32)/10) 0.7 exp(−(V + 44)/20) 1.5 lin((V + 30)/10) 0.01 exp((V + 44)/12)
40 exp(−(V + 57)/18) 10 sig((V + 14)/10) 1.875 exp(−(V + 40)/80) 0.01 exp(−(V + 44)/12)
The functions ‘sig’ and ‘lin’ are defined in Eq. (6).
The parameter values were set as Eleak = −50 mV, ENa = 55 mV, EK = −90 mV, ECa = 120 mV, gleak = 0.05 mScm2 , gNa = 45 mS/cm2 , gK = 18 mS/cm2, gNaP = 0.14 mS/cm2 , gCa = 1 mS/cm2, gKCa = 15 mS/cm2, gext,1 = 0, and gext,n = 20 nS (n = 2, · · · , 7). The parameter gM is variable. 2.4 Synapse models The EPSCs generated by presynaptic spike bursts were calculated by the following double exponential formula: Isyn = gsyn s(t − τdelay )(V(t) − EAMPA ), ds e(t) − s(t) , = dt τs de e(t) = δ(t − tspike ) − , dt τe
(8) (9) (10)
spikes
where gsyn represents the maximum conductance of an AMPA synapse, variables e and s the effective amount of released transmitters and synaptic activity, respectively. The parameter tspike referrers to the times at which spike bursts arrive at the synapse. V is postsynaptic membrane potential. The reversal potential EAMPA is 0 ms. Decay constants were set as τe = 0.2ms and τs = 5.0ms. In Fig. 1, V(t) was set to a const and τdelay = 2.5ms. 2.5 Phase reduction technique Any dynamical system which has a stable limit cycle can be reduced to a system of one dimensional phase oscillator, if the perturbation given to the system is sufficiently weak (Kuramoto 1984; Ermentrout and Kopell 1984). Using this technique, any coupled system of neurons can be reduced to a coupled system of phase oscillators, if the synaptic couplings between them are sufficiently weak. Analyzing the coupled phase oscillators, we can find stable phase differences in the steady state of a coupled neuron pair (van Vreeswijk et al. 1994; Hansel et al. 1995; Ermentrout 1996). If the coupled neuron pair exhibits in-phase synchrony as a stable state, a large-scale network of such neurons has a perfectly synchronized activity as its stable state. The reduced phase equation then takes the following general form: dφ =ω+ Z n (φ) · Isyn,n (t), dt n
(11)
where T is the period of repetitive bursting, ω = 2π/T the intrinsic bursting frequency of an uncoupled
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Here, Vn represents the single neuron’s periodic responses of the membrane potential at n-th compartment and s the synaptic activity. Since the function depends only on the phase difference between the two neurons, we can derive the equation obeyed by the relative phase ψ = φ1 − φ2 as
PRC Z
(a)
0 0
π
22
Phase φ / Time t
2π 25
d odd (ψ) < 0. dψ
0 Isyn
Interaction Γ
(b)
dψ = odd (ψ) ≡ (ψ) − (−ψ). (15) dt Each phase locked solution corresponds to a fixed point equation, odd (ψ) = 0. The equation always has two obvious solutions, ψ = 0 and ψ = π , which represent in-phase synchronous firing and anti-phase synchronous firing, respectively. A solution ψ can be reached in a steady state, if it satisfies the stability condition,
0
0
π
π Phase Difference Ψ
2π
Fig. 1 A schematic illustration of a typical phase response curve (PRC) for doublet bursting. (a) The PRC often has broad (solid) and sharp (dashed) peaks representing the responses to the first and second spikes in each burst, respectively. Two arrows indicate the times of two spike inputs from another bursting neuron during one bursting cycle. (b) We calculated the interaction function from the PRC having only the broad peak (solid curve) or only the sharp peak (dashed curve), using an EPSC induced by doublet spikes without synaptic depression (inset). We assumed a synaptic delay of 2.5 ms. When the PRC has both peaks, the corresponding interaction function is given as the sum of the two contributions (gray curve: here, α = 1)
neuron, and Z n the PRC obtained when the membrane potential of the n-th compartment is perturbed. From the above equation, we can obtain the following reduced equations for a system of the neuron pair which are reciprocally coupled with each other through the synaptic contacts at the n-th dendritic compartments: dφ1 = ω + gsyn (φ1 − φ2 ), dt dφ2 = ω + gsyn (φ2 − φ1 ), dt
(16)
In this paper, we computed PRCs by numerically integrating the so-called adjoint equation along the periodic orbit (Ermentrout 1996; Hoppensteadt and Izhikevich 1997). For instance, the equation for Izhikevich model can be given as dZ ˆ T · Z, = −dF dt
(17)
ˆ represents the Jacobian where Z = (Z v , Z u ) and dF matrix calculated from the right-hand sides of Eqs. (1) and (2). Since the PRCs are represented as unstable orbits of the adjoint equation, we should integrate the adjoint equation in the reverse direction. In so doing, spike generation induces a discontinuous change in the v-component of the PRC, Z v , of Izhikevich model (see Eq. (3)). At these discontinuous points, we may update Z v according to the following relationship: Z v (φd − 0) =
Fv (φd + 0)Z v (φd + 0) Fv (φd − 0) +
(Fu (φd + 0) − Fu (φd − 0)) Z u (φd ) , Fv (φd − 0) (18)
(12)
where φd represents a phase at which the spike generation occurs and X(φd ± 0) = limε→0 X(t ± ε). We can derive the above expression from the fact that Z · F = ω and Z u must be continuous at φ = φd .
(13)
2.6 Artificial PRCs of doublet bursting
where φ1 and φ2 represent the phase variables of the two neural oscillators, and 1 T φ (EAMPA −Vn (t))dt. (14) (φ) = Z n (ωt)s t− T0 ω
Our previous study of single-compartment CAN current model has revealed that PRCs for burst firing can in general show a broad peak in the inter-burst interval and a sharp peak in the interval within a burst (Aoyagi et al. 2003). The two peaks play differential
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(a)
193
near-singlet doublet
singlet 40 T = 22.9566
T = 28.1772
triplet
T = 23.3765
T = 25.6628
v
0
near-triplet doublet
-40
PRC Zv
-80
0 π Phase φ
0
(b)
π Phase φ
0
π Phase φ
0
π Phase φ
0
dv/dt = 0
u
150 120 90 60
-60
-30 v
0
30
-60
-30 v
0
30
-60
-30 v
0
30
-60
-30 v
0
30
Fig. 2 Dynamical behaviors and phase responses of a simple bursting neuron model. (a) The membrane potentials (upper panels) and PRCs (lower panels) during one bursting cycle were calculated in singlet, doublet near the singlet, doublet near the triplet, and triplet bursting states, respectively, using a neuron model proposed by Izhikevich (2003). (b) The corresponding trajectories are shown by thick curves in the (u, v) space. The thin parabola curve displays v-nullcline. Note that the trajectory
of the near-singlet doublet or triplet bursting passes the vicinity of v-nullcline between spikes in a burst (shaded circle). This implies that the time evolution of the intrinsic state is slow before generating the second (in doublet) or the third spike (in triplet). The critical slowing down results in a sharp peak in the PRC during intraburst spiking and in turn induces in-phase synchrony. Dashed lines represent the reset of the dynamical variables after spikes
roles in the phase-locking of burst firing. Here, we demonstrate that the sharp peak plays a crucial role in synchronization with bursts, using artificial PRCs for doublet bursting (see Fig. 1(a)). We considered the case where the interval between consecutive doublets and between two spikes in each doublet were 25 and 3 ms, respectively. Then, we described the profile of the artificial PRC in terms of one broad peak, t 3 Z B (t) = sin π (0 ≤ t ≤ 22), (19) 22
may express the entire artificial PRC for doublet bursting as Z (t) = Z B (t) + α Z S (t), where α(> 0) controls the height of the sharp peak relative to that of the broad peak. We investigate how the steady phase difference between a neuron pair would depend on the relative height of the two peaks.
and one sharp peak, t − 22 π Z S (t) = sin 3
(22 ≤ t ≤ 25).
(20)
Here, we have expressed the PRC in terms of the time variable, which defines the phase as φ = 2π t/25. We
2.7 Numerical methods Numerical computations were performed on Pentium PCs machines running with the GNU/LINUX operating system. The software for the computations was written in C++. The ordinary differential equations were integrated using the Dormand and Prince formula of order 5(4) (Dormand and Prince 1980). We obtained the exact periodic responses of neurons through the dense output method of the formula (Shampine 1986) and Newton method.
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3 Results 3.1 Impact of rhythmic bursting on PRC: general argument The PRC describes an advance or a delay induced in the rhythmic spike firing by a presynaptic spike at given phase. We have previously found that a presynaptic spike given between two spikes within a burst can be most effective for shifting the timing of subsequent output spikes, and that burst firing may promote synchrony in a coupled neuron pair showing such a response property (Aoyagi et al. 2003). Below, we explain this situation taking doublet bursting as an example. The PRC of a neuron in general expresses a broad peak in an intermediate range of the phase variable. Figure 1(a) schematically illustrates an example of such a PRC. When a neuron shows doublet spikes, another peak appears at a phase close to 2π , where 0 or 2π corresponds to the time at which the second spike is completed in each burst (see Fig. 2(a) or 7(a,c)). The height of the second peak depends crucially on the details of different neuron models or values of parameters in the same model. We calculated the interaction functions (φ) from the artificial PRC and an EPSC mediated by an AMPA synapse (inset, Fig. 1(b)). The interaction function (gray curve, Fig. 1(b)) is given as the sum of the contribution from the broad peak (solid curve, Fig. 1(b)) and that from the sharp peak in the PRC (dashed curve, Fig. 1(b)). We find that in-phase phase-locked firing is stable in the presence of the sharp peak in the PRC, whereas such a state is unstable in the absence of that peak. 3.2 Relationship between bursting dynamics and PRC We now clarify the mechanism to have the large phase response using a mathematically tractable model of bursting neuron.We employ the model proposed by Izhikevich (2003) for describing the responses of various cortical neurons (Methods). Figure 2(a) displays the temporal profiles of the membrane potential and the PRCs in different bursting modes: singlet, near-singlet doublet, near-triplet doublet, and triplet. The PRCs in the near-singlet doublet and triplet modes exhibit eminent peaks at φ ∼ 2π , whereas the others do not. The trajectories of these bursting states in the u-v space may clarify why they have different PRC shapes (Fig. 2(b)). In the near-singlet doublet mode, the trajectory is reset to the vicinity of the v-nullcline after the generation of the first spike in a doublet. The evolution of the fast variable v is very slow
J Comput Neurosci (2007) 23:189–200
in this region (in fact, a deviation from the nullcline contributes to dv/dt only through a quadratic expression, which is very small in the vicinity). In addition, the reset point defined by Eq. (3) would correspond to the saddle-node point in the continuous system without the resetting process. In general, the state approaches a saddle-node in the direction on a stable manifold, and then it escapes from the point in the direction on an unstable manifold. Since the derivatives of all dynamical variables are almost vanishing in the vicinity of the saddle-node point (circlued in Fig. 2(b)), the time evolution of the intrinsic state can be very slow. If the neuron receives a spike near this point, the input pushes the state in the direction of positive v. Because this direction is almost orthogonal to v-nullcline, the perturbed state can escape quickly from the slow trajectory along the v-nullcline. This implies that the input produces a large phase shift in the PRC. A similar situation holds for the triplet bursting between the second and third spikes. By contrast, the trajectory of the singlet firing has no time point at which the state evolution exhibits a critical slowing down. The trajectory of the near-triplet doublet does not pass the vicinity of the v-nullcline during each burst, and consequently the phase response is small during the bursting period. These results and those in Fig. 1 explain why and how a coupled pair of bursting neurons exchanges the stability of synchronous and asynchronous firing when the neurons change their intrinsic bursting mode.
3.3 FRB in the NaP current model The validity of the above neuronal dynamics view must be tested in realistic bursting neuron models. For this purpose, we first developed a NaP current-based bursting neuron model. This NaP current-based model has six dendritic compartments and is similar to the two-compartment version proposed by Wang (1999). The responses of the extended model to a constant input current are shown in Fig. 3(a). In the range of frequencies at which the model produces FRB, the response patterns resemble those of the CAN currentbased model or real FRB neurons. We can change the number of spikes per burst, or the bursting mode, by modifying the magnitude of the AHP current: if gM is decreased, more spikes are elicited in each burst at given input intensity (Fig. 3(b)). This manipulation does not significantly change the intervals between the bursts or the frequency of bursting. At a given value of gM , we can increase the frequency of bursting by increasing the intensity of external input, gext (data not
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195
(a)
g M [mS/cm 2] 1.12
0.92
0.52
0.32
30 mV
0.72
0.72
between synaptic inputs and the ping-pong mechanism of bursting on the same dendrite. Then, increasing the number of dendritic compartments may decouple these two effects and significantly change the properties of the neuron model. We analyzed the phase response of the NaP currentbased neuron model to an AMPA receptor-mediated synaptic current. Figure 4 displays the phase differences in the steady state when reciprocal AMPA synapses between a neuron pair are located on the 1st (soma), 4th or 7th (most distal dendritic) compartments, respectively. The decay constant of the synaptic current and the delay in the synaptic transmission are given as τs = 5 ms and τdelay = 1 ms, respectively. The synaptic connections located at the soma (n = 1) synchronize the neuron pair with zero phase difference
100 msec
singlet
50 40
1.1
1
0.9 0.8 0.7 2 g M [mS/cm ]
0.6
Fig. 3 Typical responses of the extended NaP current-based neuron model. The model proposed by Wang (1999) was modified to have six dendritic compartments. (a) A stepwise decrease of the maximum conductance of M-current gradually increases the number of spikes per burst (i.e., the bursting mode). The neuron eventually exhibits high-frequency tonic firing. (b) The bursting mode (upper) and bursting frequency (lower) are shown as functions of the maximum conductance of M current. The bursting frequency means an inverse of the interval between successive bursts
shown). As we increase the input intensity, the intervals between individual bursts become shorter and shorter until the bursting behavior finally turns into tonic firing. A network of the original two-compartment models showed neither in-phase nor anti-phase synchrony of bursting states, when the number of intraburst spikes was relatively small (Takekawa et al. 2004). We speculated that these results were due to interference
phase difference Ψ
phase difference Ψ
4 3 2 1
phase difference Ψ
bursting freqency [Hz] #spikes per burst
(b)
doublet stable unstable
triplet
quartet
n=1
/2
0 n=4 /2
0 n=7 /2
0 1.1
1
0.9 0.8 0.7 g M [mS/cm 2]
0.6
Fig. 4 Steady states in a coupled pair of the extended NaP current-based model. The stable phase differences (filled squares) obtained by the phase reduction method are shown for the three cases where reciprocal AMPA synapses are located on different dendritic compartments. Crosses represent unstable solutions to odd (ψ) = 0. From the top, soma (n = 1), the third (n = 4) and the most distal dendrites (n = 7). Vertical dashed bars separate the parameter regions for different bursting modes
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(a)
(c) ts
ZV
n=1 n = 4 (x 2.5) n = 7 (x 5.0)
td
V7 [mV]
0.2 0
π Phase φ
0
2π
-37
0.2mV
Unperturbed Perturbed Phase advance
ts
-38
-60 0.6 ts
0.5 0.4
π
-60
V1
-40
(d)
td
-40
(so -20 m 0 a)
20 -40
-20 V1 [mV]
0
-20 V1 [mV]
0
0 V7 [mV]
activity of M current
(b)
-37 -38 ) -39 distal t s o V7 (m
-36
-37
1mV
-38 td
-60
-40
Fig. 5 Dynamical trajectories and PRCs of the extended NaP current-based neuron model: near-singlet doublet bursting state. (a) The phase response curves were calculated with a perturbative input to the soma (n = 1) or the dendritic compartment (n = 4, 7). The PRCs of the dencritic compartments are magnified 2.5 (n = 4) or 5.0 times (n = 7), respectively. The conductance of Mcurrent was set as gM = 1. Time points ts and td refer to the times of the highest peak in the PRCs of n = 1 and n = 7, respectively. (b) The corresponding 3-dimensional dynamical trajectory is shown. Crosses display equidistance-in-time points to indicate the speed of the time evolution along the trajectory. The circled area is the slow portion of the trajectory. (c) Part of the trajectory
shown in b was magnified during bursting. At time ts , we applied a delta-function-like perturbative input to the soma with an amplitude of 0.2 mV (arrow). Note that the size of the stimulus was emphasized for the sake of visualization. Squares and circles were plotted at every 0.5 ms along the unperturbed (black) and perturbed (gray) trajectories. The perturbed trajectory escapes from the slow portion faster than the unperturbed one, and hence generates a phase advance. (d) At time td , we applied a deltafunction-like input with an amplitude of 1 mV to the most distal dendrite. The direction of this perturbation was almost parallel to the V7 axis, and the phase was advanced after the first spike
in a wide range of g M . However the coupled neurons exhibit nonzero phase differences in singlet firing, neartriplet doublet and near quartet triplet bursting states. A slower decay constant or a longer synaptic delay increased the phase differences and narrowed the domains of stable in-phase synchrony (data not shown). The synaptic connections made on the middle dendritic compartments (n = 4) induce stable anti-synchronous firing with π phase lag. Singlet firing turns to be antisynchronous, whereas both in-phase and anti-phase synchronous firing states are stable in the other burst-
ing modes. Except for the doublet bursting, the two synchronous firing states change sharply at certain values of g M . The doublet firing state exhibits the bistability in stable synchronous firing states at g M ∼ 0.9. However, this bistability is sensitive to the choice of parameter values and hence is not robust. Locating the connections at a further distal dendritic site (n = 7) makes the transitions in the stable synchronous firing state sharper and eliminates the bistability in the doublet firing. In addition, the domains of in-phase synchrony are narrower, probably because the longer
J Comput Neurosci (2007) 23:189–200
Figure 5(a) shows the phase responses of the extended NaP current-based model to perturbative stimuli given at different compartments. The PRCs can take negative values at phases near 0 or 2π . The location of the highest peak in the PRCs depends on the site of stimulation. The more distal the dendritic site of synaptic contacts, the earlier the highest peak of the PRC. Then, Fig. 5(b) displays the trajectory of a near-singlet doublet bursting state of the neuron model. As in Fig. 2(b), this trajectory has a large portion (circled) along which the state vector evolves very slowly between the two spikes within a burst. We demonstrate that small differences in the dynamical trajectories significantly vary the speed at which the state vector escapes from the vicinity of the slow portion. Below, we attempt to relate the synchronization properties of the neuron model to the corresponding bursting dynamics. To this end, we investigate the impact on the PRCs of a perturbative stimulus given to the soma at time ts or to the distal dendrite at time td (see Fig. 5(a)). Figure 5(c) shows the two-dimensional portrait of the trajectory when such a perturbative input (arrow) is applied to the soma. The perturbation is perpendicular to the tangential direction of the trajectory. The perturbed trajectory (gray curve) starts leaving the unperturbed one (black curve) at the slow portion (circled) of the trajectory, and takes a shorter route to the resting portion. This short cut results in a large phase advance in the PRC between the two spikes during a burst (see the peak at time ts in Fig. 5(b)). In Fig. 5(d), we applied a perturbative input (arrow) to the most distal dendrite at time td . Note that the amplitude of the dendritic stimulation is about five times larger than that of the stimulus to the soma, so that the two stimuli may have nearly equal impacts at the some. Since the dendritic perturbation is in the tangential direction of the trajectory, it advances the phase of the perturbed trajectory (gray curve, Fig. 5(b)). Moreover, the phase advance is further increased in the vicinity of the slow portion. The corresponding PRC, however, exhibits only a small peak for a perturbation to the dendrite, when it is as weak as the somatic perturbation. In contrast to these results, the input produces no large phase advance in the corresponding PRC between two spikes of burst (solid and dashed, Fig. 6(a)). Indeed the trajectory of the near-triplet doublet bursting does not have a large slow portion between the two spikes in
0.2 ZV
3.4 View of the PRCs in the NaP current-based bursting mechanism
(a) 0
n=1 n = 4 (x 2.5) n = 7 (x 5.0)
-0.2 0
(b) activity of M current
dendritic transmission to the soma creates an effect similar to the synaptic delay.
197
π Phase φ
2π
0.6 0.5
0
π
0.4 -60 V1
-40 (so -20 m 0 a)
20 -40
-37 -38 -39 al) t dist s o m V7 (
-36
Fig. 6 Dynamical trajectories and PRCs of the extended NaP current-based neuron model: near-triplet doublet bursting state. (a) The PRCs were calculated with a perturbative input to the soma (n = 1) or the dendrite (n = 4, 7). The PRCs of the dencritic compartments are magnified 2.5 (n = 4) or 5.0 times (n = 7), respectively. The M-current conductance was gM = 0.8. None of the PRCs has clearly distinct peaks. (b) The corresponding three-dimensional dynamical trajectory is shown. No obvious slow portion exists along the trajectory to prevent the bursting state from exhibiting in-phase synchrony
the doublet (Fig. 6(b)). Therefore, a perturbative input given between the two spikes produces no shortcut of the trajectory, irrespective of the site of stimulation. Thus, just as in the previous simple neuron model, we can interpret the properties of synchronization in the NaP current-based bursting neuron through those of the dynamical trajectories. 3.5 FRB in the CAN current-based model We have previously proposed a single-compartment model of chattering neuron based on the CAN current. A coupled pair of this neuron model displayed a synchronization switching property that is quite similar to the one explained above with the simple bursting neuron model (Aoyagi et al. 2003). We do not repeat to show these results here. Instead, we demonstrate that the basic mechanism underlying the switching behavior of the CAN current-based model, which remained unclear in our previous PRC analysis, can be understood
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(a)
(c) V [mV]
40
0 -40
0 -40
0.4
0.4
0.2
0.2
ZV
ZV
V [mV]
40
0
0 π Phase φ
0
2π
(b)
π Phase φ
0
2π
(d) 0.3
0.3
0.2
0.1
0 -60
m KCa
m KCa
0 π
0.2
0
π
0.1
-40
-20 0 V [mV]
20
40
-60
-40
-20 0 V [mV]
20
40
Fig. 7 Differences from a dynamical viewpoint between the near-singlet and near-triplet doublet bursting states in the CAN current-based neuron model. (a, b) The PRC and dynamical trajectory are shown for a near-singlet doublet bursting state. In
the trajectory, the state vector moves slowly in the vicinity of the slow portion between two spikes in the doublet (circled). (c, d) The PRC and dynamical trajectory are shown for a near-triplet doublet bursting state. The trajectory has only a tiny slow portion
by means of the dynamic trajectory such as shown in Fig. 2. Figure 7(a) shows the firing patterns and the PRCs of the near-singlet and near-triplet doublet bursting states. The two response patterns quite resemble one another. Nevertheless, a neuron pair show in-phase or anti-phase phase-locked firing in the former or latter bursting mode, respectively (Aoyagi et al. 2003). Figure 7(b) presents the dynamical trajectory projected onto the two-dimensional space spanned by a fast variable, V, and a slow variable, m KCa . The trajectory of the near-singlet doublet state has a relatively large portion (circled) along which the state evolves only slowly between two spikes. Therefore, a perturbative input given at this portion can produce a large phase advance in the
second spike of the burst. By contrast, such a portion is smaller in the trajectory of the near-triplet doublet, so the corresponding PRC acquires no sharp positive peak between the two spikes in a burst. Thus, the situation is similar to what we encountered in the dou blet bursting states of the previous two models.
4 Discussion We have shown that all three neuron models generating FRB exhibit sharp transitions between in-phase and anti-phase phase-locked activities. The exchanges in the stable phase difference accompany the changes in the bursting mode, suggesting a tight relationship
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between the intrinsic and network-level neuronal dynamics in a broad class of the biological mechanisms of bursting. We have demonstrated that the exchanges between in-phase and anti-phase phase-locked states are governed by a sharp peak in the PRC during each burst (Fig. 1), and that whether a particular burst firing generates such a peak depends crucially on the expression of a slow portion within the bursting part of the dynamical trajectory (Figs. 5, 6, 7). We have explained why this rule holds by using a mathematically tractable bursting neuron model (Izhikevich 2003) (Fig. 2). Several factors, such as the bursting frequency, the conduction delay or the decay constant of excitatory synaptic current may influence the synchronization properties to various degrees (Crook et al. 1997, 1998; Buzsaki and Draguhn 2004). These effects were not demonstrated in this paper except the case where shortterm synaptic depression enhances the stability of inphase synchronous doublet bursting (Fig. 1). In short, the in-phase phase-locked state tends to be more stable if the frequency of bursting is lower or if the synaptic decay constant is shorter. In our extended NaP current-based model, the parameter ranges that ensure the stability of in-phase synchrony are wider when the reciprocal synaptic contacts are located on more proximal dendritic compartments. By contrast, the stability exchanges between in-phase and anti-phase phase-locked states occur more sharply if the synaptic contacts are made on more distal dendritic compartments (see Fig. 4). We could successfully interpret these differences in the network behavior by inspecting the expressions of the slow portion in the bursting part of the trajectory. In summary, we have revealed the general mechanism that relates the stability exchanges between different phase-locked firing states to the exchanges in the bursting modes of individual neurons. Our results have explained how the relationship holds in a wide class of bursting neuron models and provided a simple criterion to judge whether a coupled pair of a particular bursting neuron type may exhibit the in-phase phaselocked state. Acknowledgements This work was partially supported by Grant-in-Aid for Scientific Researches of Priority Areas No.17022036 from Japanese Ministry of Education, Culture, Sports, Science and Technology.
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199 Aoyagi, T., Takekawa, T., & Fukai, T. (2003). Gamma rhythmic bursts: Coherence control in networks of cortical pyramidal neurons. Neural Computation, 15, 1035–1061. Booth, V., & Bose, A. (2002). Burst synchrony patterns in hippocampal pyramidal cell model netowkrks. Network: Computation in Neural Systems, 13, 157–177. Brumberg, J. C., Nowak, L. G., & McCormick, D. A. (2000). Ionic mechanisms underlynig repetitive high-frequency burst firing in supragranular corticalneurons. Journal of Neuroscience, 20, 4829–4843. Buzsaki, G., & Draguhn, A. (2004). Neuronal oscillations in cortical networks. Science, 304, 1926–1929. Cardin, J. A., Palmer, L. A., & Contreras, D. (2005). Stimulusdependent gamma (30–50 Hz) oscillations in simple and complex fast rhythmic bursting cells in primary visual cortex. Journal of Neuroscience, 25, 5339–5350. Crook, S. M., Ermentrout, G. B., & Bower, J. M. (1998). Dendritic and synaptic effects in systems of coupled cortical oscillators. Journal of Computational Neuroscience, 5, 515–329. Crook, S. M., Ermentrout, G. B., & Vanier, M. C. (1997). The role of axonal delay in the synchronization of networks of coupled cortical oscillators. Journal of Computational Neuroscience, 4, 161–172. Cunningham, M. O., Whittington, M. A., Bibbig, A., Roopun, A., LeBeau, F. E. N., Vogt, A., et al. (2004). A role for fast rhythmic bursting neurons in cortical gamma oscillations in vitro. Proceedings of the National Academy of Sciences of the United States of America, 101, 7152–7157. Doiron, B., Laing, C., Longtin, A., & Maler, L. (2002). Ghostbursting: A novel neuronal burst mechanism. Journal of Computational Neuroscience, 12, 5–25. Dormand, J., & Prince, P. (1980). A family of embedded rungekutta formulae. Journal of Computational and Applied Mathematics, 6, 19–26. Ermentrout, G. B. (1996). Type I membranes, phase resetting curves, and synchrony. Neural Computation, 8, 979– 1001. Ermentrout, G. B., & Kopell, N. (1984). Frequency plateaus in a chain of weakly coupled oscillators. SIAM journal on Mathematical Analysis, 15, 215–237. Goldberg, J. A., Deister, C. A., & Wilson, C. J. (2007). Response properties and synchronization of rhythmically firing dendritic neurons. Journal of Neurophysiology, 97, 208–219. Gray, C. M., & McCormick, D. A. (1996). Chattering cells: Superficial pyramidal neurons contributing to the generation of synchronous oscillations in the visual cortex. Science, 274, 109–113. Gray, C. M., & Prisco, G. V. D. (1997). Stimulus-dependent neuronal oscillations and local synchronization in striate cortex of the alert cat. Journal of Neuroscience, 17, 3239– 3253. Gray, C. M., & Singer, W. (1989). Stimulus-specific neuronal oscillations in orientation columns of cat visua-cortex. Proceedings of the National Academy of Sciences of the United States of America, 86, 1698–1702. Gutkin, B. S., Ermentrout, G. B., & Reyes, A. D. (2005). Phaseresponse curves give the responses of neurons to transient inputs. Journal of Neurophysiology, 94, 1623–1635. Haj-Dahmane, S., & Andrade, R. (1997). Calcium-activated cation nonselective currenct contributes to the fast afterdepolarization in rat prefrontal cortex neurons. Journal of Neurophysiology, 78, 1983–1989. Hansel, D., Mato, G., & Meunier, C. (1995). Synchrony in excitatory neural network. Neural Computation, 7, 307–337. Hoppensteadt, F. C., & Izhikevich, E. M. (1997). Weakly connected neural networks. New York: Springer-Verlag.
200 Izhikevich, E. (2000). Neural excitability, spiking and bursting. International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 10, 1171–1266. Izhikevich, E. M. (2003). Simple model of spiking neurons. IEEE Transactions on Neural Networks, 14, 1569–1572. Kang, Y., & Kayano, F. (1994). Electrophysiological and morphological characteristics of layer VI pyramidal cells in the cat motor cortex. Journal of Neurophysiology, 72, 578– 91. Kang, Y., Okada, T., & Ohmori, H. (1998) A phenytoin-sensitive cationic current participates in generating afterdepolarization adn burst afterdischarge in rat neocortical pyramidal cells. European Journal of Neuroscience, 10, 1363– 1375. Kuramoto, Y. (1984). Chemical oscillations, waves, and turbulence. Berlin: Springer-Verlag. Netoff, T. I., Banks, M. I., Dorval, A. D., Acker, C. D., Haas, J. S., Kopell, N., et al. (2005). Synchronization in hybrid neuronal networks of the hippocampal formation. Journal of Neurophysiology, 93, 1197–1208. Pinsky, P. F., & Rinzel, J. (1994). Intrinsic and network rhythmogenesis in a reduced traub model for CA3 neurons. Journal of Computational Neuroscience, 1, 39–60. Reyes, A. D., & Fetz, E. E. (1993). Two modes of interspike interval shortening by brief transient depolarizations in cat neocortical neurons. Journal of Neurophysiology, 69, 1661– 1672. Shampine, L. F. (1986). Some practical Runge–Kutta formulas. Mathematics of Computation, 46, 135–150. Singer, W. (1999). Neuronal synchrony a versatile code for the definition of relations? Neuron, 24, 49–65. Steriade, M. (2006). Grouping of brain rhythms in corticothalamic systems. Neuroscience, 137, 1087–1106.
J Comput Neurosci (2007) 23:189–200 Steriade, M., Timofeev, I., Dürmüller, N., & Grenier, F. (1998). Dynamic properties of corticothalamic neurons and local cortical interneurons generation fast rhythmic (30–40 Hz) spike bursts. Journal of Neurophysiology, 79, 483–490. Takekawa, T., Aoyagi, T., & Fukai, T. (2004). Influences of synaptic location on the synchronization of rhythmic bursting neurons. Network: Computation in Neural Systems, 15, 1–12. Tallon-Baudry, C., Bertrand, O., Delpuech, C., & Pernier, J. (1997). Oscillatory gamma-band (30–70 Hz) activity induced by a visual search task in humans. Journal of Neuroscience, 17, 722–734. Traub, R. D., Buhl, E. H., Gloveli, T., & Whittington, M. A. (2003). Fast rhythmic bursting can be induced in layer 2/3 corical neurons by enhancing persisten Na+ conductance or by blocking BK channels. Journal of Neurophysiology, 89, 909–921. Tsubo, Y., Takada, M., & Fukai, T. (2005). Layer-specific synchronization properties and their cholinergic modulations of rat motor cortex pyramidal neurons. In Abstract of Annual Meeting of the Society for Neuroscience, Washington DC (p 971.13). van Vreeswijk, C., Abbott, L. F., & Ermentrout, G. B. (1994). When inhibition not excitation synchronizes neuronal firing. Journal of Computational Neuroscience, 1, 313–321. Wang, X. J. (1999). Fast burst firing and short-term synaptic plasticity: A model of neocortical chattering neurons. Neuroscience, 89, 347–362. Wilson, G. F., Richardson, F. C., Fisher, T. E., Olivera, B. M., & Kaczmarek, L. K. (1996). Identification and characterization of a Ca2+ -sensitive nonspecific cation channel underlying prolonged repetitive firing in aplysia neurons. Journal of Neuroscience, 16, 3661–3671.
Synchronous and asynchronous bursting states: role of intrinsic neural dynamics Takashi Takekawa · Toshio Aoyagi · Tomoki Fukai
Received: 8 March 2006 / Revised: 28 December 2006 / Accepted: 20 February 2007 / Published online: 27 March 2007 © Springer Science + Business Media, LLC 2007
Abstract Brain signals such as local field potentials often display gamma-band oscillations (30–70 Hz) in a variety of cognitive tasks. These oscillatory activities possibly reflect synchronization of cell assemblies that are engaged in a cognitive function. A type of pyramidal neurons, i.e., chattering neurons, show fast rhythmic bursting (FRB) in the gamma frequency range, and may play an active role in generating the gamma-band oscillations in the cerebral cortex. Our previous phase response analyses have revealed that the synchronization between the coupled bursting neurons significantly depends on the bursting mode that is defined as the number of spikes in each burst. Namely, a network of neurons bursting through a Ca2+ -dependent mechanism exhibited sharp transitions between synchronous and asynchronous firing states when the neurons exchanged the bursting mode between singlet, doublet and so on. However, whether a broad class of bursting neuron models commonly show such a network behavior remains unclear. Here, we analyze the mechanism underlying this network behavior using a mathematically tractable neuron model. Then we extend our results to a multi-compartment version of the NaP current-based neuron model and prove a similar tight
Action Editor: Xiao-Jing Wang T. Takekawa (B) · T. Fukai Laboratory for Neural Circuit Theory, RIKEN Brain Science Institute, 2-1 Hirosawa, Wako, Saitama 351-0198 Japan e-mail: [email protected] T. Aoyagi Graduate School of Informatics, Kyoto University, Yoshida-Honmachi, Sakyo, 606-8501 Kyoto, Japan
relationship between the bursting mode changes and the network state changes in this model. Thus, the synchronization behavior couples tightly to the bursting mode in a wide class of networks of bursting neurons. Keywords Stimulus-dependent synchronization · Fast rhythmic burst · Phase response · Synaptic location
1 Introduction A type of pyramidal neurons called “fast rhythmic bursting (FRB)” or “chattering” neuron responds to a visual stimulus or a depolarizing step current with rhythmic bursts of spikes in the gamma frequency range (30–70 Hz) (Gray and McCormick 1996; Steriade et al. 1998; Cardin et al. 2005). These neurons are distinguished from other types of cortical neurons showing burst firing by their rather high-intraburst firing frequencies (> 200–300 Hz) (Gray and McCormick 1996), and may promote synchronization of neuronal activity in local cortical areas (Gray and Singer 1989; TallonBaudry et al. 1997; Gray and Prisco 1997; Singer 1999; Cunningham et al. 2004; Steriade 2006). The phase response curve (PRC) describes how a perturbative input to a neuron advances or delays a postsynaptic spike response in a rhythmically firing state (Kuramoto 1984; Ermentrout and Kopell 1984), and has recently been measured in cortical neurons (Reyes and Fetz 1993; Gutkin et al. 2005; Netoff et al. 2005; Tsubo et al. 2005; Goldberg et al. 2007). We have previously shown using PRCs that synchronization or desynchronization in a network of rhythmically bursting neurons might depend on the intrinsic mechanism
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of bursting. Some experimental evidence suggested that the calcium-activated non-specific cation (CAN) current (Wilson et al. 1996; Haj-Dahmane and Andrade 1997) is essential for FRB (Kang and Kayano 1994; Kang et al. 1998; Aoyagi et al. 2002). Other studies suggested that the persistent sodium (NaP) current and the electronic interplay between soma and dendrite are crucial for FRB (Wang 1999; Brumberg et al. 2000; Traub et al. 2003). Therefore we analyzed the synchronization properties of two types of neuron models when the same type of neurons were coupled via AMPA receptor-mediated excitatory synapses (Aoyagi et al. 2003; Takekawa et al. 2004). The coupled neuron system of the CAN current-based single-compartment model underwent an exchange of the stability from anti-phase (with π phase lag) to in-phase (with 0 phase lag) synchronous firing when the bursting mode, or the number of spikes per burst, increased (Aoyagi et al. 2003). By contrast, the NaP current-based twocompartment neuron model showed neither in-phase nor anti-phase phase-locked state in many cases, and the stability exchanges between them were also less evident (Takekawa et al. 2004). These results raise several questions about the relationship between the network dynamics for synchronization and the bursting dynamics which is intrinsic to cell types (Pinsky and Rinzel 1994; Izhikevich 2000; Booth and Bose 2002; Doiron et al. 2002). Are the correlated changes between the bursting mode and the stability of synchronous firing a general property of bursting neuron models? How does the intrinsic mechanism of bursting influence the network dynamics of synchronization? Here, we address these issues by introducing a coupled system of simple bursting neuron models (Izhikevich 2003). We then increase the number of dendritic compartments in the NaP current-based two-compartment neuron model to show that the tight relationship between the burst mode and synchronous firing holds for the extended model.
where x(t ± 0) = limε→0 x(t ± ε). Here, the values of the parameters were set as a = 0.02, b = 0.2, c = −50, d = 80, 50, 39, 28 (singlet, two types of doublet, and triplet), and I = 35. To make the sizes of u and v the same order, we have rescaled the variables in the original model as u ← u/b and d ← d/b . 2.2 Single-compartment CAN current model In the present paper, we analyzed two different realistic pyramidal neuron models that exhibit rhythmic bursting in the gamma frequency range (20–80 Hz): a single-compartment model based on Ca2+ -activated cation (CAN) current and a multi-compartment model based on NaP current. The CAN current-based model is described by the following equation for the membrane potential: Cm
dV = −Ileak − INa − IDK − ICa dt −IKCa − ICAN + Iext ,
(4)
where Ileak , INa and IDK are a leak current and spikegenerating sodium and potassium currents; ICa , IKCa and ICAN represent high-voltage-activated Ca2+ current, Ca2+ -activated potassium current (SK current) and Ca2+ -activated cation current, respectively; and Iext is an external input current. Since the mathematical details of this model have been published elsewhere (Aoyagi et al. 2002, 2003), below we list only the parameter values that were different from the previous ones: T = 309.16, KCAN = 13.16 μM and gCAN = 113 or 170 (two types of doublet) mS/cm2 . 2.3 Multi-compartment NaP current model
2 Materials and methods 2.1 Izhikevich model We used the bursting neuron model constructed by Izhikevich (2003). The model is described by twodimensional ordinary differential equations of the following form: dv = 0.04v + 5v 2 + 140 − b u + I ≡ Fv (u, v), dt du = a(v − u) ≡ Fu (u, v). dt
At each postsynaptic spike, the variables are reset as follows: v(t + 0) = c if v(t − 0) = 30 then (3) u(t + 0) = u(t − 0) + d,
(1) (2)
We extended the previously proposed twocompartment neuron model (Wang 1999). The present model has one somatic compartment and six dendritic compartments. The membrane potential of the n-th compartment Vn obeys the following equation:
Cm
dVn Vn − Vn−1 Vn − Vn+1 = − In − + dt Rn−1 An Rn An −
Iext,n + Isyn,n An
(n = 1, 2, · · · , 7),
(5)
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191
where n represents the serial position of the compartment from the soma which is indexed as n = 1. The membrane capacitance Cm is 1 μF/cm2 . The somatic compartment’s area A1 is 4950 μm2 and the dendritic compartment’s areas An (n = 2, · · · , 7) are fixed at 4675 μm2 . The axial resistance between adjacent compartments Rn (n = 1, · · · , 6) are fixed at 10 M. In 0 8 the above equations, we set as V1R−V = 0 and V7R−V =0 0 7 at both ends. The somatic compartment includes a leak current, the spike-generating sodium and potassium currents, and muscarine-dependent potassium current (M-current), while the dendritic compartments include a leak current, persistent sodium current, highvoltage-activated Ca2+ current, and Ca2+ -activated potassium current: I1 = Ileak + INa + IDK + IM ; In = Ileak + INaP + ICa + IKCa (n = 2, · · · , 7). These ionic currents are represented as Ileak = gleak (V − Eleak ), INa = gNa m3 h(V − ENa ), IK = gK m4 (V − EK ), IM = gM m(V − EK ), INaP = gNaP sig((V + 45)/5)(V − ENa ), ICa = gCa sig((V + 20)/10)(V − ECa ) and IKCa = gKCa [Ca2+ ]i (V − EK ), where [Ca2+ ] +30 i
sig(x) ≡
1 , 1 + exp(−x)
lin(x) ≡
x . 1 − exp(−x)
(6)
The activation (m) and inactivation (h) variables of each ionic current obey the following equation: dx/dt = α(1 − x) − βx. The coefficients α and β may depend on the membrane potential. The explicit forms of these functions are listed in Table 1 for the types of ionic current used in the present models. The concentration of intracellular calcium [Ca2+ ]i obeys the following differential equation: d[Ca2+ ]i = −0.002ICa − 0.005[Ca2+ ]i . dt
(7)
We investigated the neuronal responses when an external current was applied to different compartments. We consider the case that the input current is mediated by AMPA glutamate receptors: Iext = gext (V − EAMPA ). We change the intensity of the input current by changing the value of gext . Table 1 The coefficent functions of the activation and inactivation variables for the ionic currents of the NaP current model Type
α
β
Na, m Na, h DK, m M, m
10 lin((V + 32)/10) 0.7 exp(−(V + 44)/20) 1.5 lin((V + 30)/10) 0.01 exp((V + 44)/12)
40 exp(−(V + 57)/18) 10 sig((V + 14)/10) 1.875 exp(−(V + 40)/80) 0.01 exp(−(V + 44)/12)
The functions ‘sig’ and ‘lin’ are defined in Eq. (6).
The parameter values were set as Eleak = −50 mV, ENa = 55 mV, EK = −90 mV, ECa = 120 mV, gleak = 0.05 mScm2 , gNa = 45 mS/cm2 , gK = 18 mS/cm2, gNaP = 0.14 mS/cm2 , gCa = 1 mS/cm2, gKCa = 15 mS/cm2, gext,1 = 0, and gext,n = 20 nS (n = 2, · · · , 7). The parameter gM is variable. 2.4 Synapse models The EPSCs generated by presynaptic spike bursts were calculated by the following double exponential formula: Isyn = gsyn s(t − τdelay )(V(t) − EAMPA ), ds e(t) − s(t) , = dt τs de e(t) = δ(t − tspike ) − , dt τe
(8) (9) (10)
spikes
where gsyn represents the maximum conductance of an AMPA synapse, variables e and s the effective amount of released transmitters and synaptic activity, respectively. The parameter tspike referrers to the times at which spike bursts arrive at the synapse. V is postsynaptic membrane potential. The reversal potential EAMPA is 0 ms. Decay constants were set as τe = 0.2ms and τs = 5.0ms. In Fig. 1, V(t) was set to a const and τdelay = 2.5ms. 2.5 Phase reduction technique Any dynamical system which has a stable limit cycle can be reduced to a system of one dimensional phase oscillator, if the perturbation given to the system is sufficiently weak (Kuramoto 1984; Ermentrout and Kopell 1984). Using this technique, any coupled system of neurons can be reduced to a coupled system of phase oscillators, if the synaptic couplings between them are sufficiently weak. Analyzing the coupled phase oscillators, we can find stable phase differences in the steady state of a coupled neuron pair (van Vreeswijk et al. 1994; Hansel et al. 1995; Ermentrout 1996). If the coupled neuron pair exhibits in-phase synchrony as a stable state, a large-scale network of such neurons has a perfectly synchronized activity as its stable state. The reduced phase equation then takes the following general form: dφ =ω+ Z n (φ) · Isyn,n (t), dt n
(11)
where T is the period of repetitive bursting, ω = 2π/T the intrinsic bursting frequency of an uncoupled
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Here, Vn represents the single neuron’s periodic responses of the membrane potential at n-th compartment and s the synaptic activity. Since the function depends only on the phase difference between the two neurons, we can derive the equation obeyed by the relative phase ψ = φ1 − φ2 as
PRC Z
(a)
0 0
π
22
Phase φ / Time t
2π 25
d odd (ψ) < 0. dψ
0 Isyn
Interaction Γ
(b)
dψ = odd (ψ) ≡ (ψ) − (−ψ). (15) dt Each phase locked solution corresponds to a fixed point equation, odd (ψ) = 0. The equation always has two obvious solutions, ψ = 0 and ψ = π , which represent in-phase synchronous firing and anti-phase synchronous firing, respectively. A solution ψ can be reached in a steady state, if it satisfies the stability condition,
0
0
π
π Phase Difference Ψ
2π
Fig. 1 A schematic illustration of a typical phase response curve (PRC) for doublet bursting. (a) The PRC often has broad (solid) and sharp (dashed) peaks representing the responses to the first and second spikes in each burst, respectively. Two arrows indicate the times of two spike inputs from another bursting neuron during one bursting cycle. (b) We calculated the interaction function from the PRC having only the broad peak (solid curve) or only the sharp peak (dashed curve), using an EPSC induced by doublet spikes without synaptic depression (inset). We assumed a synaptic delay of 2.5 ms. When the PRC has both peaks, the corresponding interaction function is given as the sum of the two contributions (gray curve: here, α = 1)
neuron, and Z n the PRC obtained when the membrane potential of the n-th compartment is perturbed. From the above equation, we can obtain the following reduced equations for a system of the neuron pair which are reciprocally coupled with each other through the synaptic contacts at the n-th dendritic compartments: dφ1 = ω + gsyn (φ1 − φ2 ), dt dφ2 = ω + gsyn (φ2 − φ1 ), dt
(16)
In this paper, we computed PRCs by numerically integrating the so-called adjoint equation along the periodic orbit (Ermentrout 1996; Hoppensteadt and Izhikevich 1997). For instance, the equation for Izhikevich model can be given as dZ ˆ T · Z, = −dF dt
(17)
ˆ represents the Jacobian where Z = (Z v , Z u ) and dF matrix calculated from the right-hand sides of Eqs. (1) and (2). Since the PRCs are represented as unstable orbits of the adjoint equation, we should integrate the adjoint equation in the reverse direction. In so doing, spike generation induces a discontinuous change in the v-component of the PRC, Z v , of Izhikevich model (see Eq. (3)). At these discontinuous points, we may update Z v according to the following relationship: Z v (φd − 0) =
Fv (φd + 0)Z v (φd + 0) Fv (φd − 0) +
(Fu (φd + 0) − Fu (φd − 0)) Z u (φd ) , Fv (φd − 0) (18)
(12)
where φd represents a phase at which the spike generation occurs and X(φd ± 0) = limε→0 X(t ± ε). We can derive the above expression from the fact that Z · F = ω and Z u must be continuous at φ = φd .
(13)
2.6 Artificial PRCs of doublet bursting
where φ1 and φ2 represent the phase variables of the two neural oscillators, and 1 T φ (EAMPA −Vn (t))dt. (14) (φ) = Z n (ωt)s t− T0 ω
Our previous study of single-compartment CAN current model has revealed that PRCs for burst firing can in general show a broad peak in the inter-burst interval and a sharp peak in the interval within a burst (Aoyagi et al. 2003). The two peaks play differential
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(a)
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near-singlet doublet
singlet 40 T = 22.9566
T = 28.1772
triplet
T = 23.3765
T = 25.6628
v
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-80
0 π Phase φ
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u
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-60
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Fig. 2 Dynamical behaviors and phase responses of a simple bursting neuron model. (a) The membrane potentials (upper panels) and PRCs (lower panels) during one bursting cycle were calculated in singlet, doublet near the singlet, doublet near the triplet, and triplet bursting states, respectively, using a neuron model proposed by Izhikevich (2003). (b) The corresponding trajectories are shown by thick curves in the (u, v) space. The thin parabola curve displays v-nullcline. Note that the trajectory
of the near-singlet doublet or triplet bursting passes the vicinity of v-nullcline between spikes in a burst (shaded circle). This implies that the time evolution of the intrinsic state is slow before generating the second (in doublet) or the third spike (in triplet). The critical slowing down results in a sharp peak in the PRC during intraburst spiking and in turn induces in-phase synchrony. Dashed lines represent the reset of the dynamical variables after spikes
roles in the phase-locking of burst firing. Here, we demonstrate that the sharp peak plays a crucial role in synchronization with bursts, using artificial PRCs for doublet bursting (see Fig. 1(a)). We considered the case where the interval between consecutive doublets and between two spikes in each doublet were 25 and 3 ms, respectively. Then, we described the profile of the artificial PRC in terms of one broad peak, t 3 Z B (t) = sin π (0 ≤ t ≤ 22), (19) 22
may express the entire artificial PRC for doublet bursting as Z (t) = Z B (t) + α Z S (t), where α(> 0) controls the height of the sharp peak relative to that of the broad peak. We investigate how the steady phase difference between a neuron pair would depend on the relative height of the two peaks.
and one sharp peak, t − 22 π Z S (t) = sin 3
(22 ≤ t ≤ 25).
(20)
Here, we have expressed the PRC in terms of the time variable, which defines the phase as φ = 2π t/25. We
2.7 Numerical methods Numerical computations were performed on Pentium PCs machines running with the GNU/LINUX operating system. The software for the computations was written in C++. The ordinary differential equations were integrated using the Dormand and Prince formula of order 5(4) (Dormand and Prince 1980). We obtained the exact periodic responses of neurons through the dense output method of the formula (Shampine 1986) and Newton method.
194
3 Results 3.1 Impact of rhythmic bursting on PRC: general argument The PRC describes an advance or a delay induced in the rhythmic spike firing by a presynaptic spike at given phase. We have previously found that a presynaptic spike given between two spikes within a burst can be most effective for shifting the timing of subsequent output spikes, and that burst firing may promote synchrony in a coupled neuron pair showing such a response property (Aoyagi et al. 2003). Below, we explain this situation taking doublet bursting as an example. The PRC of a neuron in general expresses a broad peak in an intermediate range of the phase variable. Figure 1(a) schematically illustrates an example of such a PRC. When a neuron shows doublet spikes, another peak appears at a phase close to 2π , where 0 or 2π corresponds to the time at which the second spike is completed in each burst (see Fig. 2(a) or 7(a,c)). The height of the second peak depends crucially on the details of different neuron models or values of parameters in the same model. We calculated the interaction functions (φ) from the artificial PRC and an EPSC mediated by an AMPA synapse (inset, Fig. 1(b)). The interaction function (gray curve, Fig. 1(b)) is given as the sum of the contribution from the broad peak (solid curve, Fig. 1(b)) and that from the sharp peak in the PRC (dashed curve, Fig. 1(b)). We find that in-phase phase-locked firing is stable in the presence of the sharp peak in the PRC, whereas such a state is unstable in the absence of that peak. 3.2 Relationship between bursting dynamics and PRC We now clarify the mechanism to have the large phase response using a mathematically tractable model of bursting neuron.We employ the model proposed by Izhikevich (2003) for describing the responses of various cortical neurons (Methods). Figure 2(a) displays the temporal profiles of the membrane potential and the PRCs in different bursting modes: singlet, near-singlet doublet, near-triplet doublet, and triplet. The PRCs in the near-singlet doublet and triplet modes exhibit eminent peaks at φ ∼ 2π , whereas the others do not. The trajectories of these bursting states in the u-v space may clarify why they have different PRC shapes (Fig. 2(b)). In the near-singlet doublet mode, the trajectory is reset to the vicinity of the v-nullcline after the generation of the first spike in a doublet. The evolution of the fast variable v is very slow
J Comput Neurosci (2007) 23:189–200
in this region (in fact, a deviation from the nullcline contributes to dv/dt only through a quadratic expression, which is very small in the vicinity). In addition, the reset point defined by Eq. (3) would correspond to the saddle-node point in the continuous system without the resetting process. In general, the state approaches a saddle-node in the direction on a stable manifold, and then it escapes from the point in the direction on an unstable manifold. Since the derivatives of all dynamical variables are almost vanishing in the vicinity of the saddle-node point (circlued in Fig. 2(b)), the time evolution of the intrinsic state can be very slow. If the neuron receives a spike near this point, the input pushes the state in the direction of positive v. Because this direction is almost orthogonal to v-nullcline, the perturbed state can escape quickly from the slow trajectory along the v-nullcline. This implies that the input produces a large phase shift in the PRC. A similar situation holds for the triplet bursting between the second and third spikes. By contrast, the trajectory of the singlet firing has no time point at which the state evolution exhibits a critical slowing down. The trajectory of the near-triplet doublet does not pass the vicinity of the v-nullcline during each burst, and consequently the phase response is small during the bursting period. These results and those in Fig. 1 explain why and how a coupled pair of bursting neurons exchanges the stability of synchronous and asynchronous firing when the neurons change their intrinsic bursting mode.
3.3 FRB in the NaP current model The validity of the above neuronal dynamics view must be tested in realistic bursting neuron models. For this purpose, we first developed a NaP current-based bursting neuron model. This NaP current-based model has six dendritic compartments and is similar to the two-compartment version proposed by Wang (1999). The responses of the extended model to a constant input current are shown in Fig. 3(a). In the range of frequencies at which the model produces FRB, the response patterns resemble those of the CAN currentbased model or real FRB neurons. We can change the number of spikes per burst, or the bursting mode, by modifying the magnitude of the AHP current: if gM is decreased, more spikes are elicited in each burst at given input intensity (Fig. 3(b)). This manipulation does not significantly change the intervals between the bursts or the frequency of bursting. At a given value of gM , we can increase the frequency of bursting by increasing the intensity of external input, gext (data not
J Comput Neurosci (2007) 23:189–200
195
(a)
g M [mS/cm 2] 1.12
0.92
0.52
0.32
30 mV
0.72
0.72
between synaptic inputs and the ping-pong mechanism of bursting on the same dendrite. Then, increasing the number of dendritic compartments may decouple these two effects and significantly change the properties of the neuron model. We analyzed the phase response of the NaP currentbased neuron model to an AMPA receptor-mediated synaptic current. Figure 4 displays the phase differences in the steady state when reciprocal AMPA synapses between a neuron pair are located on the 1st (soma), 4th or 7th (most distal dendritic) compartments, respectively. The decay constant of the synaptic current and the delay in the synaptic transmission are given as τs = 5 ms and τdelay = 1 ms, respectively. The synaptic connections located at the soma (n = 1) synchronize the neuron pair with zero phase difference
100 msec
singlet
50 40
1.1
1
0.9 0.8 0.7 2 g M [mS/cm ]
0.6
Fig. 3 Typical responses of the extended NaP current-based neuron model. The model proposed by Wang (1999) was modified to have six dendritic compartments. (a) A stepwise decrease of the maximum conductance of M-current gradually increases the number of spikes per burst (i.e., the bursting mode). The neuron eventually exhibits high-frequency tonic firing. (b) The bursting mode (upper) and bursting frequency (lower) are shown as functions of the maximum conductance of M current. The bursting frequency means an inverse of the interval between successive bursts
shown). As we increase the input intensity, the intervals between individual bursts become shorter and shorter until the bursting behavior finally turns into tonic firing. A network of the original two-compartment models showed neither in-phase nor anti-phase synchrony of bursting states, when the number of intraburst spikes was relatively small (Takekawa et al. 2004). We speculated that these results were due to interference
phase difference Ψ
phase difference Ψ
4 3 2 1
phase difference Ψ
bursting freqency [Hz] #spikes per burst
(b)
doublet stable unstable
triplet
quartet
n=1
/2
0 n=4 /2
0 n=7 /2
0 1.1
1
0.9 0.8 0.7 g M [mS/cm 2]
0.6
Fig. 4 Steady states in a coupled pair of the extended NaP current-based model. The stable phase differences (filled squares) obtained by the phase reduction method are shown for the three cases where reciprocal AMPA synapses are located on different dendritic compartments. Crosses represent unstable solutions to odd (ψ) = 0. From the top, soma (n = 1), the third (n = 4) and the most distal dendrites (n = 7). Vertical dashed bars separate the parameter regions for different bursting modes
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(a)
(c) ts
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ts
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-37 -38 ) -39 distal t s o V7 (m
-36
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1mV
-38 td
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-40
Fig. 5 Dynamical trajectories and PRCs of the extended NaP current-based neuron model: near-singlet doublet bursting state. (a) The phase response curves were calculated with a perturbative input to the soma (n = 1) or the dendritic compartment (n = 4, 7). The PRCs of the dencritic compartments are magnified 2.5 (n = 4) or 5.0 times (n = 7), respectively. The conductance of Mcurrent was set as gM = 1. Time points ts and td refer to the times of the highest peak in the PRCs of n = 1 and n = 7, respectively. (b) The corresponding 3-dimensional dynamical trajectory is shown. Crosses display equidistance-in-time points to indicate the speed of the time evolution along the trajectory. The circled area is the slow portion of the trajectory. (c) Part of the trajectory
shown in b was magnified during bursting. At time ts , we applied a delta-function-like perturbative input to the soma with an amplitude of 0.2 mV (arrow). Note that the size of the stimulus was emphasized for the sake of visualization. Squares and circles were plotted at every 0.5 ms along the unperturbed (black) and perturbed (gray) trajectories. The perturbed trajectory escapes from the slow portion faster than the unperturbed one, and hence generates a phase advance. (d) At time td , we applied a deltafunction-like input with an amplitude of 1 mV to the most distal dendrite. The direction of this perturbation was almost parallel to the V7 axis, and the phase was advanced after the first spike
in a wide range of g M . However the coupled neurons exhibit nonzero phase differences in singlet firing, neartriplet doublet and near quartet triplet bursting states. A slower decay constant or a longer synaptic delay increased the phase differences and narrowed the domains of stable in-phase synchrony (data not shown). The synaptic connections made on the middle dendritic compartments (n = 4) induce stable anti-synchronous firing with π phase lag. Singlet firing turns to be antisynchronous, whereas both in-phase and anti-phase synchronous firing states are stable in the other burst-
ing modes. Except for the doublet bursting, the two synchronous firing states change sharply at certain values of g M . The doublet firing state exhibits the bistability in stable synchronous firing states at g M ∼ 0.9. However, this bistability is sensitive to the choice of parameter values and hence is not robust. Locating the connections at a further distal dendritic site (n = 7) makes the transitions in the stable synchronous firing state sharper and eliminates the bistability in the doublet firing. In addition, the domains of in-phase synchrony are narrower, probably because the longer
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Figure 5(a) shows the phase responses of the extended NaP current-based model to perturbative stimuli given at different compartments. The PRCs can take negative values at phases near 0 or 2π . The location of the highest peak in the PRCs depends on the site of stimulation. The more distal the dendritic site of synaptic contacts, the earlier the highest peak of the PRC. Then, Fig. 5(b) displays the trajectory of a near-singlet doublet bursting state of the neuron model. As in Fig. 2(b), this trajectory has a large portion (circled) along which the state vector evolves very slowly between the two spikes within a burst. We demonstrate that small differences in the dynamical trajectories significantly vary the speed at which the state vector escapes from the vicinity of the slow portion. Below, we attempt to relate the synchronization properties of the neuron model to the corresponding bursting dynamics. To this end, we investigate the impact on the PRCs of a perturbative stimulus given to the soma at time ts or to the distal dendrite at time td (see Fig. 5(a)). Figure 5(c) shows the two-dimensional portrait of the trajectory when such a perturbative input (arrow) is applied to the soma. The perturbation is perpendicular to the tangential direction of the trajectory. The perturbed trajectory (gray curve) starts leaving the unperturbed one (black curve) at the slow portion (circled) of the trajectory, and takes a shorter route to the resting portion. This short cut results in a large phase advance in the PRC between the two spikes during a burst (see the peak at time ts in Fig. 5(b)). In Fig. 5(d), we applied a perturbative input (arrow) to the most distal dendrite at time td . Note that the amplitude of the dendritic stimulation is about five times larger than that of the stimulus to the soma, so that the two stimuli may have nearly equal impacts at the some. Since the dendritic perturbation is in the tangential direction of the trajectory, it advances the phase of the perturbed trajectory (gray curve, Fig. 5(b)). Moreover, the phase advance is further increased in the vicinity of the slow portion. The corresponding PRC, however, exhibits only a small peak for a perturbation to the dendrite, when it is as weak as the somatic perturbation. In contrast to these results, the input produces no large phase advance in the corresponding PRC between two spikes of burst (solid and dashed, Fig. 6(a)). Indeed the trajectory of the near-triplet doublet bursting does not have a large slow portion between the two spikes in
0.2 ZV
3.4 View of the PRCs in the NaP current-based bursting mechanism
(a) 0
n=1 n = 4 (x 2.5) n = 7 (x 5.0)
-0.2 0
(b) activity of M current
dendritic transmission to the soma creates an effect similar to the synaptic delay.
197
π Phase φ
2π
0.6 0.5
0
π
0.4 -60 V1
-40 (so -20 m 0 a)
20 -40
-37 -38 -39 al) t dist s o m V7 (
-36
Fig. 6 Dynamical trajectories and PRCs of the extended NaP current-based neuron model: near-triplet doublet bursting state. (a) The PRCs were calculated with a perturbative input to the soma (n = 1) or the dendrite (n = 4, 7). The PRCs of the dencritic compartments are magnified 2.5 (n = 4) or 5.0 times (n = 7), respectively. The M-current conductance was gM = 0.8. None of the PRCs has clearly distinct peaks. (b) The corresponding three-dimensional dynamical trajectory is shown. No obvious slow portion exists along the trajectory to prevent the bursting state from exhibiting in-phase synchrony
the doublet (Fig. 6(b)). Therefore, a perturbative input given between the two spikes produces no shortcut of the trajectory, irrespective of the site of stimulation. Thus, just as in the previous simple neuron model, we can interpret the properties of synchronization in the NaP current-based bursting neuron through those of the dynamical trajectories. 3.5 FRB in the CAN current-based model We have previously proposed a single-compartment model of chattering neuron based on the CAN current. A coupled pair of this neuron model displayed a synchronization switching property that is quite similar to the one explained above with the simple bursting neuron model (Aoyagi et al. 2003). We do not repeat to show these results here. Instead, we demonstrate that the basic mechanism underlying the switching behavior of the CAN current-based model, which remained unclear in our previous PRC analysis, can be understood
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(c) V [mV]
40
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0 π
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40
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20
40
Fig. 7 Differences from a dynamical viewpoint between the near-singlet and near-triplet doublet bursting states in the CAN current-based neuron model. (a, b) The PRC and dynamical trajectory are shown for a near-singlet doublet bursting state. In
the trajectory, the state vector moves slowly in the vicinity of the slow portion between two spikes in the doublet (circled). (c, d) The PRC and dynamical trajectory are shown for a near-triplet doublet bursting state. The trajectory has only a tiny slow portion
by means of the dynamic trajectory such as shown in Fig. 2. Figure 7(a) shows the firing patterns and the PRCs of the near-singlet and near-triplet doublet bursting states. The two response patterns quite resemble one another. Nevertheless, a neuron pair show in-phase or anti-phase phase-locked firing in the former or latter bursting mode, respectively (Aoyagi et al. 2003). Figure 7(b) presents the dynamical trajectory projected onto the two-dimensional space spanned by a fast variable, V, and a slow variable, m KCa . The trajectory of the near-singlet doublet state has a relatively large portion (circled) along which the state evolves only slowly between two spikes. Therefore, a perturbative input given at this portion can produce a large phase advance in the
second spike of the burst. By contrast, such a portion is smaller in the trajectory of the near-triplet doublet, so the corresponding PRC acquires no sharp positive peak between the two spikes in a burst. Thus, the situation is similar to what we encountered in the dou blet bursting states of the previous two models.
4 Discussion We have shown that all three neuron models generating FRB exhibit sharp transitions between in-phase and anti-phase phase-locked activities. The exchanges in the stable phase difference accompany the changes in the bursting mode, suggesting a tight relationship
J Comput Neurosci (2007) 23:189–200
between the intrinsic and network-level neuronal dynamics in a broad class of the biological mechanisms of bursting. We have demonstrated that the exchanges between in-phase and anti-phase phase-locked states are governed by a sharp peak in the PRC during each burst (Fig. 1), and that whether a particular burst firing generates such a peak depends crucially on the expression of a slow portion within the bursting part of the dynamical trajectory (Figs. 5, 6, 7). We have explained why this rule holds by using a mathematically tractable bursting neuron model (Izhikevich 2003) (Fig. 2). Several factors, such as the bursting frequency, the conduction delay or the decay constant of excitatory synaptic current may influence the synchronization properties to various degrees (Crook et al. 1997, 1998; Buzsaki and Draguhn 2004). These effects were not demonstrated in this paper except the case where shortterm synaptic depression enhances the stability of inphase synchronous doublet bursting (Fig. 1). In short, the in-phase phase-locked state tends to be more stable if the frequency of bursting is lower or if the synaptic decay constant is shorter. In our extended NaP current-based model, the parameter ranges that ensure the stability of in-phase synchrony are wider when the reciprocal synaptic contacts are located on more proximal dendritic compartments. By contrast, the stability exchanges between in-phase and anti-phase phase-locked states occur more sharply if the synaptic contacts are made on more distal dendritic compartments (see Fig. 4). We could successfully interpret these differences in the network behavior by inspecting the expressions of the slow portion in the bursting part of the trajectory. In summary, we have revealed the general mechanism that relates the stability exchanges between different phase-locked firing states to the exchanges in the bursting modes of individual neurons. Our results have explained how the relationship holds in a wide class of bursting neuron models and provided a simple criterion to judge whether a coupled pair of a particular bursting neuron type may exhibit the in-phase phaselocked state. Acknowledgements This work was partially supported by Grant-in-Aid for Scientific Researches of Priority Areas No.17022036 from Japanese Ministry of Education, Culture, Sports, Science and Technology.
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